MSc: Advanced Statistics

Advanced Statistics

• Course name: Advanced Statistics
• Course number: DS-03

Course Characteristics

Key concepts of the class

• Statistical inference
• Non parametric statistics
• Test of statistical hypotheses
• Simple linear regression and correlation analysis
• Meta-Analysis

What is the purpose of this course?

The main purpose of this course is to present the fundamentals of inferential statistics to the future software engineers and data scientists, on one side providing the scientific fundamentals of the disciplines, and on the other anchoring the theoretical concepts on practices coming from the world of software development and engineering. The course covers the statistical analysis of data with limited assumptions on the distribution, with reference to testing hypotheses, measuring correlations, building samples, and performing regressions.

- What should a student remember at the end of the course?

By the end of the course, the students should be able to:

• Remember the fundamentals of inferential statistics
• Remember the specifics and purpose of different hypothesis tests
• Distinguish between parametric and non parametric tests

- What should a student be able to understand at the end of the course?

By the end of the course, the students should be able to understand:

• the basic concepts of inferential statistics
• the fundamental laws in statistics
• the concept of null and alternative hypotheses
• the hypotheses test procedure

- What should a student be able to apply at the end of the course?

By the end of the course, the students should be able to ...

• To understand the problems related to analyse statistically data not distributed normally
• To know the more recent computationally-intensive techniques that can help to describe samples and to infer properties of populations in absence of normality
• To identify situations when the data is on nominal scales so alternative techniques should be use, and act accordingly.
• To be able to run experiment to evaluate hypotheses for situation of scarce data, distributed non normally, on different kinds of scales.

Course evaluation

Course grade breakdown

		Proposed points
Weekly quizzes	?	10
Midterm	?	20
Final oral exam	?	35
Final written exam	?	30
Participation	?	5

Grades range

Course grading range

		Proposed range
A. Excellent	90-100	95-100
B. Good	75-89	75-94
C. Satisfactory	60-74	55-74
D. Poor	0-59	0-54

Resources and reference material

• Wasserman L. (2006) All of Nonparametric Statistics. Springer
• Randles, R.H. and Wolfe, D.A. (1991). Introduction to the Theory of Nonparametric Statistics. Melbourne: Robert Krieger. (Ch.1‐Ch.4)
• Hastie, T. Tibshirani, R. and Friedman, J. (2008) The Elements of Statistical Learning 2ed. Springer
• Hollander, M. and Wolfe, D.A. (1999). Nonparametric Statistical Methods, 2nd ed. New York: John Wiley.

Course Sections

The main sections of the course and approximate hour distribution between them is as follows:

Course Sections

Section	Section Title	Teaching Hours
1	Sampling Distributions Associated with the Normal Population	15
2	Test of Statistical Hypotheses	30
3	Simple Linear Regression and Correlation Analysis	15

Section 1

Section title:

Sampling Distributions Associated with the Normal Population

Topics covered in this section:

• Introduction to the course, toward inference
• Student’s t-distribution
• Bernoulli and binomial distribution
• Chi-square distribution
• Snedecor’s F-distribution

What forms of evaluation were used to test students’ performance in this section?

|a|c| & Yes/No
Development of individual parts of software product code & 0
Homework and group projects & 0
Midterm evaluation & 1
Testing (written or computer based) & 1
Reports & 0
Essays & 0
Oral polls & 0
Discussions & 1

Typical questions for ongoing performance evaluation within this section

1. Deduce the probability mass function ${\textstyle P(X\leq k}$ for a binomial distribution?
2. Let X1,...,Xk be k iid random variables distributed with a ${\textstyle \chi ^{2}}$ distribution with n1,...nk degrees of freedom respectively.
   

What is the distribution of Y=X1+...+Xk? Define it precisely and prove the answer formally?

1. List at least 3 random variables that “tend to follow” a t distribution?
2. If X has Chi square function with the 5 degrees of freedom, then what is the probability that X is between 1.145 and 12.83?
3. If X has a gamma distribution of (1,1), then what is the probability density function of the random variable 2X?

Typical questions for seminar classes (labs) within this section

1. Define and provide examples of sample space, events and probability measure.
2. Write the formula for the coefficients of the simple linear regression. Explain the mathematical procedure you do to derive them and derive them.
3. Calculate the correlation between two functions and explain its meaning.
4. Calculate the Pearson coefficient for the given functions.
5. Deduce the MGF for normal distribution.
6. State and prove the Bonferroni inequality.

Section 2

Section title:

Test of Statistical Hypotheses

Topics covered in this section:

• Z-test
• Student’s t-test
• Chi-square test
• Snedecor’s F-test

What forms of evaluation were used to test students’ performance in this section?

|a|c| & Yes/No
Development of individual parts of software product code & 0
Homework and group projects & 0
Midterm evaluation & 1
Testing (written or computer based) & 1
Reports & 0
Essays & 0
Oral polls & 0
Discussions & 1

Typical questions for ongoing performance evaluation within this section

1. Define the concept of power of a statistical test.
2. Define the purpose of the F Test, its hypotheses, and its structure.
3. Define the purpose of the t-Test, its hypotheses, and its structure.
4. Define the purpose of the Chi square Test, its hypotheses, and its structure.
5. Define the purpose of the Z Test, its hypotheses, and its structure.
6. Provide concrete numeric examples with explanation on why the power of a test depends on:
   1. the size of the data sets.
   2. the magnitude of the effect.
   3. the level of statistical significance.
7. Given a statistical test for which we have set a value ${\textstyle \alpha }$ we obtain a p:
   1. if we can reject H0 ${\textstyle (p<\alpha )}$, what we typically say about H0 and H1.
   2. if we cannot reject H0 ${\textstyle (P\geq \alpha )}$, what we can typically say about H0 and H1.
   3. when can we say that H0 holds?
   4. when can we say that H1 holds?

Typical questions for seminar classes (labs) within this section

1. Provide a concrete example of a t test, detailing both H0 and H1.
2. Present the structure of the F test for the analysis of the variance.
3. Explain what are H0 and H1 in hypothesis testing.
4. Explain the role of the Bonferroni inequality in hypothesis testing.

Section 3

Section title:

Simple Linear Regression and Correlation Analysis

Topics covered in this section:

• Kolmogorov-Smirnov test
• Size of samples, Kolmogorov-Smirnov, Fisher exact
• Logistic regression

What forms of evaluation were used to test students’ performance in this section?

|a|c| & Yes/No
Development of individual parts of software product code & 0
Homework and group projects & 0
Midterm evaluation & 0
Testing (written or computer based) & 1
Reports & 0
Essays & 0
Oral polls & 0
Discussions & 1

Typical questions for ongoing performance evaluation within this section

1. Let X1,X2, ...,X10 be a random sample from a distribution whose probability density function is ${\textstyle f(x)=(1if0<x<1}$, otherwise 0). Based on the observed values 0.62, 0.36, 0.23, 0.76, 0.65, 0.09, 0.55, 0.26, 0.38, 0.24, test the hypothesis H0 : X UNIF(0, 1) against H1 : X UNIF(0, 1) at a significance level = 0.1.
2. If X1,X2, ...,Xn is a random sample from a distribution with density function ${\textstyle f(x)=((1-\theta )x^{\theta }if0<x<1}$, otherwise 0), what is the maximum likelihood estimator of ${\textstyle \theta }$?
3. Let X1,X2, ...,Xn be a random sample of size n from a distribution with a probability density function ${\textstyle f(x)=((1-\theta )x^{\theta }if0<x<1,}$ otherwise 0), where ${\textstyle 0<\theta }$ is a parameter. Using the maximum likelihood method find an estimator for the parameter ${\textstyle \theta }$.
4. Suppose you are told that the likelihood of ${\textstyle \theta }$ at ${\textstyle \theta =2}$ is given by 1/4. Is this the probability that ${\textstyle \theta =2}$? Explain why or why not.

Typical questions for seminar classes (labs) within this section

1. If X1,X2, ...,Xn is a random sample from a distribution with density function${\textstyle f(x)=({\frac {1}{\theta }}if0<x<1,}$ otherwise 0), then what is the maximum likelihood estimator of ${\textstyle \theta }$?
2. Let X1,X2, ...,Xn be a random sample from a normal population with mean ${\textstyle \mu }$ and variance ${\textstyle \sigma ^{2}}$. What are the maximum likelihood estimators of ${\textstyle \mu }$ and ${\textstyle \sigma ^{2}}$?
3. Suppose that you have the following data points: 0.36, 0.32, 0.10, 0.13, 0.45, 0.11, 0.12, 0.09; compute Dn to determine if they come from the uniform distribution [0,0.5].
4. The data on the heights of 12 infants are given below: 18.2, 21.4, 22.6, 17.4, 17.6, 16.7, 17.1, 21.4, 20.1, 17.9, 16.8, 23.1. Test the hypothesis that the data came from some normal population at a significance level = 0.1.

Test questions for final assessment in the course

1. Providing full example of two sequences (in case of computational overhead, you can approximate at the first decimal digit). Compute their:
   1. Covariance.
   2. Pearson’s correlation coefficient.
   3. Spearman’s Rank Correlation Coefficient.
   4. Kendall’s tau Correlation coefficient.
2. What is an empirical distribution?
3. Present, prove, and discuss the evaluation of the asymptotic confidence interval for the empirical distribution, detailing the role of the binomial.
4. Prove, under the simplified hypotheses, the distribution free property of Dn.
5. Write the Shannon Theorem and discuss its implications.
6. Discuss how we could proceed to compute the confidence interval of the Kendall Tau correlation coefficient of the population.
7. Suppose that you have the following datapoints: 0.4, 2, 0.6, 2.4, 2.2, 3.6, 3.8, 4; compute Dn to determine if they come from the uniform distribution [0,4].
8. Prove that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Tilde{F}_n} is a consistent and unbiased estimator of F.